Question

1. a student is using coordinate geometry to find the perimeter of the triangle.

image of triangle with points l(-a, b), t(a, b), m(0, c), w(0, b)
which equation could be used to find the length of segment lm?
a. $sqrt{(-a - a)^2 + (b - b)^2}$
b. $\frac{c - b}{0 - a}$
c. $\frac{b - c}{-a - 0}$
d. $sqrt{(0 + a)^2 + (c - b)^2}$

Explanation:

Step1: Recall Distance Formula

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

Step2: Identify Coordinates of L and M

Point \(L\) has coordinates \((-a, b)\) and point \(M\) has coordinates \((0, c)\).

Step3: Substitute into Distance Formula

Substitute \(x_1=-a\), \(y_1 = b\), \(x_2 = 0\), \(y_2=c\) into the distance formula:
\[

$$\begin{align*} d&=\sqrt{(0 - (-a))^2+(c - b)^2}\\ &=\sqrt{(0 + a)^2+(c - b)^2} \end{align*}$$

\]

Step4: Analyze Other Options

• Option A: The coordinates used are incorrect (uses \(T\)'s \(x\)-coordinate instead of \(M\)'s).
• Option B and C: These are slope formulas, not distance formulas, so they can't be used to find length.

Answer:

D. \(\sqrt{(0 + a)^2+(c - b)^2}\)